Spectrum RepresentationThe Spectrum of a Sum of SinusoidsA Notation ChangeBeat NotesBeat Note SpectrumBeat Note WaveformMultiplication of SinusoidsAmplitude ModulationPeriodic WaveformsNonperiodic SignalsFourier SeriesFourier Series: AnalysisFourier Series DerivationOrthogonality PropertySummarySpectrum of the Fourier SeriesFourier Analysis of Periodic SignalsThe Square WaveSpectrum for a Square WaveSynthesis of a Square WaveTriangle WaveTriangle Wave SpectrumSynthesis of a Triangle WaveConvergence of Fourier SeriesTime–Frequency SpectrumStepped FrequencySpectrogram AnalysisFrequency Modulation: Chirp SignalsChirped or Linearly Swept FrequencySummaryECE 2610 Signal and Systems 3–1Spectrum Representation• Extending the investigation of Chapter 2, we now considersignals/waveforms that are composed of multiple sinusoidshaving different amplitudes, frequencies, and phases(3.1)where here is real, is complex, and is the frequency in Hz• We desire a graphical representation of the parameters in(3.1) versus frequencyThe Spectrum of a Sum of Sinusoids• An alternative form of (3.1), which involves the use of theinverse Euler formula’s, is to expand each real cosine intotwo complex exponentials(3.2)xt A0Ak2fkt k+cosk 1=N+=X0ReXkej2fktk 1=N+=X0A0=XkAkejk=fkxt X0Xk2-----ej2fktXk*2------ej– 2fkt+k 1=N+=Chapter3The Spectrum of a Sum of SinusoidsECE 2610 Signals and Systems 3–2– Note that we now have each real sinusoid expressed as asum of positive and negative frequency complex sinusoidsTwo-Sided Sinusoidal Signal Spectrum: Express as in(3.2) and then the spectrum is the set of frequency/amplitudepairs(3.3)• The spectrum can be plotted as vertical lines along a fre-quency axis, with height being the magnitude of each orthe angle (phase), thus creating either a two-sided magnitudeor phase spectral plot, respectively– The text first introduces this plot as a combination of mag-nitude and phase, but later uses distinct plotsExample: Constant + Two Real Sinusoids(3.4)• We expand into complex sinusoid pairs(3.5)xt0 X0f1X12f–1X1*2  fkXk2f–kXk*2fNXN2f–NXN*2Xkxt 53 2 50 t  8+cos+= 62 300 t  2+cos+xtxt 532---ej 250t8---+32---ej– 250t8---+++= 62---ej 2300t2---+62---ej– 2300t2---+++The Spectrum of a Sum of SinusoidsECE 2610 Signals and Systems 3–3• The frequency pairs that define the two-sided line spectrumare(3.6)• We can now plot the magnitude phase spectra, in this casewith the help of a MATLAB custom functionfunction Line_Spectra(fk,Xk,mode,linetype)% Line_Spectra(fk,Xk,range,linetype)%% Plot Two-sided Line Spectra for Real Signals%----------------------------------------------------% fk = vector of real sinusoid frequencies% Xk = magnitude and phase at each positive frequency in fk% mode = 'mag' => a magnitude plot, 'phase' => a phase % plot in radians % linetype = line type per MATLAB definitions%% Mark Wickert, September 2006; modified February 2009 if nargin < 4 linetype = 'b';end my_linewidth = 2.0; switch lower(mode) % not case sensitive case {'mag','magnitude'} % two choices work k = 1; if fk(k) == 0 plot([fk(k) fk(k)],[0 abs(Xk(k))],linetype,... 'LineWidth',my_linewidth); hold on else Xk(k) = Xk(k)/2; plot([fk(k) fk(k)],[0 abs(Xk(k))],linetype,...0550 1.5ej 850– 1.5ej–  8,300 3ej 2300 3ej–  2–The Spectrum of a Sum of SinusoidsECE 2610 Signals and Systems 3–4 'LineWidth',my_linewidth); hold on plot([-fk(k) -fk(k)],[0 abs(Xk(k))],linetype,... 'LineWidth',my_linewidth); end for k=2:length(fk) if fk(k) == 0 plot([fk(k) fk(k)],[0 abs(Xk(k))],linetype,... 'LineWidth',my_linewidth); else Xk(k) = Xk(k)/2; plot([fk(k) fk(k)],[0 abs(Xk(k))],linetype,... 'LineWidth',my_linewidth); plot([-fk(k) -fk(k)],[0 abs(Xk(k))],linetype,... 'LineWidth',my_linewidth); end end grid axis([-1.2*max(fk) 1.2*max(fk) 0 1.05*max(abs(Xk))]) ylabel('Magnitude') xlabel('Frequency (Hz)') hold off case 'phase' k = 1; if fk(k) == 0 plot([fk(k) fk(k)],[0 angle(Xk(k))],linetype,... 'LineWidth',my_linewidth); hold on else plot([fk(k) fk(k)],[0 angle(Xk(k))],linetype,... 'LineWidth',my_linewidth); plot([-fk(k) -fk(k)],[0 -angle(Xk(k))],linetype,... 'LineWidth',my_linewidth); hold on end for k=2:length(fk) if fk(k) == 0 plot([fk(k) fk(k)],[0 angle(Xk(k))],linetype,... 'LineWidth',my_linewidth); elseThe Spectrum of a Sum of SinusoidsECE 2610 Signals and Systems 3–5 plot([fk(k) fk(k)],[0 angle(Xk(k))],linetype,... 'LineWidth',my_linewidth); plot([-fk(k) -fk(k)],[0 -angle(Xk(k))],... linetype,'LineWidth',my_linewidth); end end grid plot(1.2*[-max(fk) max(fk)], [0 0],'k'); axis([-1.2*max(fk) 1.2*max(fk) -1.1*max(abs(angle(Xk))) 1.1*max(abs(angle(Xk)))]) ylabel('Phase (rad)') xlabel('Frequency (Hz)') hold off otherwise error('mode must be mag or phase')end• We use the above function to plot magnitude and phase spec-tra for ; Note for the Xk’s we actually enter >> Line_Spectra([0 50 300],[5 3*exp(j*pi/8) 6*exp(j*pi/2)],'mag')>> Line_Spectra([0 50 300],[5 3*exp(j*pi/8) 6*exp(j*pi/2)],'phase')xtAkejk−300 −200 −100 0 100 200 30000.511.522.533.544.55MagnitudeFrequency (Hz)Line SpectrumMagnitude PlotX0a0=X12------a1=The Spectrum of a Sum of SinusoidsECE 2610 Signals and Systems 3–6A Notation Change• The conversion to frequency/amplitude pairs is a bit cumber-some since the factor of must be carried for all termsexcept ,